of a Number of ink Vibrations. 505 



that followed in a recent paper*. By taking logarithms we 

 find 



where 



V— 156- 7 '°=-3o!sr • • • <"> 



Retaining for the moment only the leading term, we get 

 $X*)d<j\a = — - ^cos {2a^a)e- n ^\lf 



= ^(t>\nir)e-" <T2na \lcrla (16) 



In comparing this with (5), v/e must observe that there x 

 denotes t.ie mean of the distances of which a is the sum, so 

 t:iat <r = nx, and thus the two results are in agreement. 



If we denote the leading term in cj) n by <I>, we obtain 

 from (13) and (14) 



d 2 & a cV<& d 4 <$ 



^ = d>4 6 2 «A 4 - -, -6hih" + i 6 V7i 4 2 ^, (17) 

 Tn die an* an v 



by means of which the approximation in powers of 1/n can 

 be pursued. The terms written would suffice for a result 

 correct to 1/n 2 inclusive, but we may content ourselves with 

 the term which is of the order 1/n in comparison with the 



leading term. We have 



<D= A / — I* - 



■6<r 2 /na- 



V w 



d 2 ® _ /(_±\ -W na* J 3 18<T 2 36^n 



dn 2 ~ V Wv) e 14 na* n?a* J ' 



and accordingly 



Here <j> (<r)da/a expresses the probability that the sum of 

 the distances, measured from the centre of the line, shall lie 

 between a and a -{-da. 



In terms of the mean (.r) of the distances, we should have 



yO'—'I'-iG-";"- 1 ^*)}''* #> 



as the probability that x shall lie between x and x + dx. It 

 * Phil. Mag. vol. xxxvii. p. 344 (1919), equations (65), (66), &c. 



